To determine whether there is an association between shirt size and hat size for the children on the baseball team, we need to carefully analyze the given data in the conditional relative frequency table.
Here's the table for reference:
| | Child-sized Hat | Adult-sized Hat | Total |
|--------------------|-----------------|-----------------|-------|
| Medium Shirt | ≈ 0.67 | ≈ 0.33 | 1.0 |
| Large Shirt | 0.2 | 0.8 | 1.0 |
| Total | 0.48 | 0.52 | 1.0 |
To identify if there is an association between shirt size and hat size, we should compare the probabilities within each shirt size row separately.
### Medium Shirt Row Analysis
- The conditional probability of wearing a child-sized hat given the child has a medium shirt is ≈ 0.67.
- The conditional probability of wearing an adult-sized hat given the child has a medium shirt is ≈ 0.33.
### Large Shirt Row Analysis
- The conditional probability of wearing a child-sized hat given the child has a large shirt is 0.2.
- The conditional probability of wearing an adult-sized hat given the child has a large shirt is 0.8.
Considering these probabilities:
- For children with medium shirts, there is a higher likelihood (≈ 0.67) to wear child-sized hats and a lower likelihood (≈ 0.33) to wear adult-sized hats.
- Conversely, for children with large shirts, the likelihood to wear adult-sized hats is quite high (0.8) and low for child-sized hats (0.2).
The stark contrast between these probabilities suggests an association between the shirt size and the hat size. Specifically:
- A large shirt size is strongly associated with wearing an adult-sized hat (0.8) versus a child-sized hat (0.2).
- This noticeable difference in conditional probabilities (0.8 versus 0.2) for large shirts indicates an association.
Hence, the correct interpretation would be:
There is likely an association because 0.8 is not similar to 0.2.
